In quantum mechanics (and computation & information), **weak measurements** are a type of quantum measurement that results in an observer obtaining very little information about the system on average, but also disturbs the state very little.^{ [1] } From Busch's theorem^{ [2] } the system is necessarily disturbed by the measurement. In the literature weak measurements are also known as unsharp,^{ [3] } fuzzy,^{ [3] }^{ [4] } dull, noisy,^{ [5] } approximate, and gentle^{ [6] } measurements. Additionally weak measurements are often confused with the distinct but related concept of the weak value.^{ [7] }

Weak measurements were first thought about in the context of weak continuous measurements of quantum systems^{ [8] } (i.e. quantum filtering and quantum trajectories). The physics of continuous quantum measurements is as follows. Consider using an ancilla, e.g. a field or a current, to probe a quantum system. The interaction between the system and the probe correlates the two systems. Typically the interaction only weakly correlates the system and ancilla. (Specifically, the interaction unitary need only to be expanded to first or second order in perturbation theory.) By measuring the ancilla and then using quantum measurement theory, the state of the system conditioned on the results of the measurement can be determined. In order to obtain a strong measurement, many ancilla must be coupled and then measured. In the limit where there is a continuum of ancilla the measurement process becomes continuous in time. This process was described first by: Mensky;^{ [9] }^{ [10] } Belavkin;^{ [11] }^{ [12] } Barchielli, Lanz, Prosperi;^{ [13] } Barchielli;^{ [14] } Caves;^{ [15] }^{ [16] } Caves and Milburn.^{ [17] } Later on Howard Carmichael ^{ [18] } and Howard M. Wiseman ^{ [19] } also made important contributions to the field.

The notion of a weak measurement is often misattributed to Aharonov, Albert and Vaidman.^{ [7] } In their article they consider an example of a weak measurement (and perhaps coin the phrase "weak measurement") and use it to motivate their definition of a weak value, which they defined there for the first time.

There is no universally accepted definition of a weak measurement. One approach is to declare a weak measurement to be a generalized measurement where some or all of the Kraus operators are close to the identity.^{ [20] } The approach taken below is to interact two systems weakly and then measure one of them.^{ [21] } After detailing this approach we will illustrate it with examples.

Consider a system that starts in the quantum state and an ancilla that starts in the state , the combined initial state is . These two systems interact via the Hamiltonian , which generates the time evolutions (in units where ), where is the "interaction strength", which has units of inverse time. Assume a fixed interaction time and that is small, such that . A series expansion of in gives

Because it was only necessary to expand the unitary to a low order in perturbation theory, we call this is a weak interaction. Further, the fact that the unitary is predominately the identity operator, as and are small, implies that the state after the interaction is not radically different from the initial state. The combined state of the system after interaction is

Now we perform a measurement on the ancilla to find out about the system, this is known as an ancilla-coupled measurement. We will consider measurements in a basis (on the ancilla system) such that . The measurements action on both systems is described by the action of the projectors on the joint state . From quantum measurement theory we know the conditional state after the measurement is

where is a normalization factor for the wavefunction. Notice the ancilla system state records the outcome of the measurement. The object is an operator on the system Hilbert space and is called a Kraus operator.

With respect to the Kraus operators the post-measurement state of the combined system is

The objects are elements of what is called a POVM and must obey so that the corresponding probabilities sum to unity: . As the ancilla system is no longer correlated with the primary system, it is simply recording the outcome of the measurement, we can trace over it. Doing so gives the conditional state of the primary system alone:

which we still label by the outcome of the measurement . Indeed, these considerations allow one to derive a quantum trajectory.

We will use the canonical example of Gaussian Kraus operators given by Barchielli, Lanz, Prosperi;^{ [13] } and Caves and Milburn.^{ [17] } Take , where the position and momentum on both systems have the usual Canonical commutation relation . Take the initial wavefunction of the ancilla to have a Gaussian distribution

The position wavefunction of the ancilla is

The Kraus operators are (compared to the discussion above, we set )

while the corresponding POVM elements are

which obey . An alternative representation is often seen in the literature. Using the spectral representation of the position operator , we can write

Notice that .^{ [17] } That is, in a particular limit these operators limit to a strong measurement of position; for other values of we refer to the measurement as finite-strength; and as , we say the measurement is weak.

As stated above, Busch's theorem^{ [2] } prevents a free lunch: there can be no information gain without disturbance. However, the tradeoff between information gain and disturbance has been characterized by many authors, including Fuchs and Peres;^{ [22] } Fuchs;^{ [23] } Fuchs and Jacobs;^{ [24] } and Banaszek.^{ [25] }

Recently the information-gain–disturbance tradeoff relation has been examined in the context of what is called the "gentle-measurement lemma".^{ [6] }^{ [26] }

Since the early days it has been clear that the primary use of weak measurement would be for feedback control or adaptive measurements of quantum systems. Indeed, this motivated much of Belavkin's work, and an explicit example was given by Caves and Milburn. An early application of an adaptive weak measurements was that of Dolinar's ^{[ dead link ]} receiver,^{ [27] } which has been realized experimentally.^{ [28] }^{ [29] } Another interesting application of weak measurements is to use weak measurements followed by a unitary, possibly conditional on the weak measurement result, to synthesize other generalized measurements.^{ [20] } Wiseman and Milburn's book^{ [21] } is a good reference for many of the modern developments.

- Brun's article
^{ [1] } - Jacobs and Steck's article
^{ [30] } - Quantum Measurement Theory and its Applications, K. Jacobs (Cambridge Press, 2014) ISBN 9781107025486
- Quantum Measurement and Control, H. M. Wiseman and G. J. Milburn (Cambridge Press, 2009)
^{ [21] } - Tamir and Cohen's article
^{ [31] }

**Quantum teleportation** is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. Moreover, the sender may not know the location of the recipient, and does not know which particular quantum state will be transferred.

In quantum mechanics, the **uncertainty principle** is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, *x*, and momentum, *p*, can be predicted from initial conditions.

**Bell's theorem** proves that quantum physics is incompatible with local hidden-variable theories. It was introduced by physicist John Stewart Bell in a 1964 paper titled "On the Einstein Podolsky Rosen Paradox", referring to a 1935 thought experiment that Albert Einstein, Boris Podolsky and Nathan Rosen used to argue that quantum physics is an "incomplete" theory. By 1935, it was already recognized that the predictions of quantum physics are probabilistic. Einstein, Podolsky and Rosen presented a scenario that, in their view, indicated that quantum particles, like electrons and photons, must carry physical properties or attributes not included in quantum theory, and the uncertainties in quantum theory's predictions were due to ignorance of these properties, later termed "hidden variables". Their scenario involves a pair of widely separated physical objects, prepared in such a way that the quantum state of the pair is entangled.

In quantum mechanics, a **density matrix** is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent *mixed states*. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.

In quantum physics, a **measurement** is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.

In quantum computing and specifically the quantum circuit model of computation, a **quantum logic gate** is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.

**Quantum error correction** (**QEC**) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.

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In functional analysis and quantum measurement theory, a **positive operator-valued measure** (**POVM**) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs.

The **Born rule** is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. It was formulated by German physicist Max Born in 1926.

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In theoretical physics, **quantum nonlocality** refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not admit an interpretation in terms of a local realistic theory. Quantum nonlocality has been experimentally verified under different physical assumptions. Any physical theory that aims at superseding or replacing quantum theory should account for such experiments and therefore must also be nonlocal in this sense; quantum nonlocality is a property of the universe that is independent of our description of nature.

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The **Ghirardi–Rimini–Weber theory** (**GRW**) is a spontaneous collapse theory in quantum mechanics, proposed in 1986 by Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber.

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In quantum information theory, **mutually unbiased bases** in Hilbert space **C**^{d} are two orthonormal bases and such that the square of the magnitude of the inner product between any basis states and equals the inverse of the dimension *d*:

In quantum mechanics, and especially quantum information theory, the **purity** of a normalized quantum state is a scalar defined as

The **Koopman–von Neumann mechanics** is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.

The **quantum Fisher information** is a central quantity in quantum metrology and is the quantum analogue of the classical Fisher information. The quantum Fisher information of a state with respect to the observable is defined as

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`|journal=`

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